The Stacks project

Lemma 31.15.3. Let $X$ be a locally Noetherian scheme.

  1. Let $D \subset X$ be a locally principal closed subscheme. Let $\xi \in D$ be a generic point of an irreducible component of $D$. Then $\dim (\mathcal{O}_{X, \xi }) \leq 1$.

  2. Let $D \subset X$ be an effective Cartier divisor. Let $\xi \in D$ be a generic point of an irreducible component of $D$. Then $\dim (\mathcal{O}_{X, \xi }) = 1$.

Proof. Proof of (1). By assumption we may assume $X = \mathop{\mathrm{Spec}}(A)$ and $D = \mathop{\mathrm{Spec}}(A/(f))$ where $A$ is a Noetherian ring and $f \in A$. Let $\xi $ correspond to the prime ideal $\mathfrak p \subset A$. The assumption that $\xi $ is a generic point of an irreducible component of $D$ signifies $\mathfrak p$ is minimal over $(f)$. Thus $\dim (A_\mathfrak p) \leq 1$ by Algebra, Lemma 10.60.11.

Proof of (2). By part (1) we see that $\dim (\mathcal{O}_{X, \xi }) \leq 1$. On the other hand, the local equation $f$ is a nonzerodivisor in $A_\mathfrak p$ by Lemma 31.13.2 which implies the dimension is at least $1$ (because there must be a prime in $A_\mathfrak p$ not containing $f$ by the elementary Algebra, Lemma 10.17.2). $\square$

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